MathDB

4

Part of 1964 Dutch Mathematical Olympiad

Problems(1)

g {g(x) } = g(x)

Source: Netherlands - Dutch NMO 1964 p4

1/31/2023
The function ƒƒ is defined at [0,1][0,1], and f{f(x)}=ƒ(x)f\{f(x)\} = ƒ(x).
∃c∈[0,1][f(c)=12]\exists _{c\in [0,1]} \left[f(c) =\frac12 \right] Determine f(12).f\left(\frac12 \right).
∀t∈[0,1]∃s∈[0,1][f(s)=t]\forall _{t\in [0,1]}\exists _{s\in [0,1]}[f(s) = t]. Determine ff.
Prove that the function gg, with g(x)=xg(x) = x,0≤x≤k0 \le x \le k, g(x)=kg(x) = k, k≤x≤1k \le x \le 1 satisfies the relation g{g(x)}=g(x)g\{g(x)\} = g(x).
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